Method and arrangement for designing a technical system

ABSTRACT

In a method for designing a technical system, a technical system is modeled by a predetermined quantity of target functions depending on parameters, each individual target function being weighted with a weighting factor. The method solves a system of equations comprising the parameters and the weighting factors as variables in a variable space, solutions of the system of equations forming working points of a solution space in the variable space. The working points are determined by a predictor-corrector method, according to which a predictor produced by a stochastic variable is determined in the variable space, from a first working point, and a second working point is then determined in a correcting step. The determined working points are used to design the technical system. The method can be used to redesign, modify or adapt an already existing technical system.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is based on and hereby claims priority to PCTApplication No. PCT/DE2003/002566 filed on Jul. 30, 2003 and GermanApplication No. 102 37 335.3 filed on Aug. 14, 2002, the contents ofwhich are hereby incorporated by reference.

BACKGROUND OF THE INVENTION

The invention relates to a method and an arrangement for designing atechnical system.

In order to design a complex technical system it is often necessary tooptimize the system with respect to a plurality of contradictorycriteria. The criteria influence target functions of the system, suchas, for example, manufacturing costs or efficiency. In addition,possible operating points of the system can be restricted by auxiliaryconditions. This leads to the problem of determining a set of optimaloperating points for the system, that is to say the set of possibleoperating points of the system with which it is not possible to optimizethe operating points further simultaneously with regard to all criteria.From the set of optimal points, individual users can then select themost suitable operating points of the system for their applicationswhile taking into account secret criteria or expert knowledge.

A weighting method for optimizing technical systems with respect to aplurality of criteria is known from C. Hillermeier: “NonlinearMultiobjective Optimization: A Generalized Homotopy Approach”, Chapter3.2, Birkhäuser Verlag, 2001 (“the Hillermeier Chapter 3.2 reference”),wherein scaling parameters are employed to apply transformations toscalar-value optimization problems. This method has the disadvantagethat it is numerically very involved, because very many scalar-valueoptimizations have to be performed. Furthermore, the selection andvariation of the scaling parameters necessitates an interaction with auser and in this respect cannot be automated.

A stochastic method for optimizing technical systems with respect to aplurality of criteria, wherein a stochastic differential equation isused to solve the optimization problem, is described in C. Hillermeier:“Nonlinear Multiobjective Optimization: A Generalized HomotopyApproach”, Chapter 3.3, Birkhäuser Verlag, 2001. This method has thedisadvantage that it are very involved in numerical terms, because amultiplicity of quadratic optimization problems have to be solved. Afurther disadvantage lies in the fact that with the method, theindividual target functions are not weighted, as a result of whichimportant information for selecting an optimal point is not available tothe user.

A homotopy method for optimizing technical systems with respect tomultiple criteria, wherein in addition to weighting factors for thetarget functions, Lagrange multipliers are used in order to takeauxiliary conditions into account, is known from C. Hillermeier: “AGeneralized Homotopy Approach to Multiobjective Optimization”, Journalof Optimization Theory and Application, Vol. 110/3, pp. 557-583, PlenumPress, New York, 2001 (“the Hillemermeier Vol. 110/3 reference”). Thedisadvantage of this method lies in the fact that an interaction withthe user is necessary and in this respect the method cannot beautomated.

SUMMARY OF THE INVENTION

One possible object of the invention is therefore to create an automatedand numerically efficient method for designing a technical system.

The inventors propose a method for designing a technical system in whichthe technical system is modeled by a predetermined set of targetfunctions which are dependent on parameters. In this modeling processeach individual target function is weighted with a weighting factor. Themethod solves an equation system comprising the parameters and theweighting factors as variables in a variable space, with solutions ofthe equation system forming operating points of a solution space in thevariable space. In the method the operating points are determined by apredictor-corrector method, according to which, starting from a firstoperating point, a predictor generated by a stochastic variable isdetermined in the variable space, and subsequently, in a corrector step,a second operating point is determined. The determined operating pointsare used here to design the technical system. The method can be used todesign a new technical system or modify or, as the case may be, adapt anexisting technical system.

An advantage resides in the fact that the method is automated throughthe generation of the predictor by a stochastic variable and so there isno longer any need for intervention on the part of the user. The linkingof the numerical predictor-corrector method with stochastic methodsguarantees efficient use of the computer resources for calculatingoperating points of a technical system.

In an advantageous embodiment the predictor is predetermined by randomnumbers, so that in particular during the execution of the method arandom number generator can be used and through this, the automation ofthe method is ensured in a simple manner.

In a further particularly advantageous embodiment the random numbers arenormally distributed. What this achieves is that the trajectory ofoperating points which forms in the solution space during the executionof the method ensures a uniform distribution in the entire solutionspace and so ensures good coverage of all possible operating points. Asa result of the use of normally distributed random numbers, inparticular a Brownian movement on the solution space can be modeled bythe method.

Preferably the operating points which are determined by the method arewhat are known as pareto-optimal points which cannot be optimizedfurther in relation to all target functions. In the method, inparticular the points with positive weighting factors in the solutionspace are selected as operating points.

In a further advantageous embodiment the operating points must alsosatisfy one or more auxiliary conditions, with the or each auxiliarycondition being represented by a further variable of the equation systemin the variable space. In this case the auxiliary conditions can beequality auxiliary conditions and/or inequality auxiliary conditions.With inequality auxiliary conditions a slack variable is preferablyintroduced, by which the inequality auxiliary conditions can betransformed into equality auxiliary conditions. The use of slackvariables will be explained in more detail in the detailed descriptionof an exemplary embodiment.

The solution space of the operating points is preferably a manifold, inparticular a submanifold in the variable space. In the HillemermeierVol. 110/3 reference it is explained under what preconditions thesolution space forms such a manifold.

Since in particular at the start of the method a first operating pointis present initially, in a special embodiment this first valid operatingpoint is determined by a weighting method, the use of weighting methodsalready being known from the related art (see the Hillermeier Chapter3.2 reference).

With the predictor-corrector method, which may be used in the invention,a tangential plane to the solution space is determined, preferably inthe first operating point, and the predictor is then specified in thetangential plane.

In a development of the method, if a negative predictor with one or morenegative weighting factors occurs, a new predictor is determined by areflection at a subplane of the solution space of the valid operatingpoints. Through this, new regions of valid operating points can bedetermined, which operating points can be of particular relevance to theuser in terms of secret supplementary criteria or his/her expertknowledge.

In a preferred embodiment, in the reflection step a point ofintersection of the trajectory that runs between the first operatingpoint and the negative predictor with a subplane of the solution spaceis determined. The tangential component of the vector spanned by thepoint of intersection and the negative predictor to the relevantsubplane of the solution space is then determined, with those weightingfactors which were negative for the negative predictor in the points ofthe subplane now being equal to zero. Next, the normal component,associated with the tangential component, of the vector spanned by thepoint of intersection and the negative predictor is determined. Finally,the new predictor is determined by two times subtraction of the normalcomponent from the negative predictor.

A Newton method known from the related art, which method is easilyconvertible numerically, is preferably used for the corrector method.

The operating points are preferably determined by iterations of thepredictor-corrector method, with the second operating point of thepreceding iteration step being used in a current iteration step as thefirst operating point of the predictor-corrector method. In this casethe method is terminated by, for example, an abort condition. In anadvantageous embodiment the abort condition is met when a predeterminednumber of operating points has been determined and/or a predeterminedtime limit has been reached.

In addition to the above-described method for designing a technicalsystem, the inventors propose an arrangement for designing a technicalsystem by which the above-described method can be performed. Inparticular the method comprises a processor unit by which it is madepossible for the predictor to be generated using a stochastic variable.

The arrangement preferably comprises a random number generator forgenerating random numbers which represent the stochastic variable.

The inventors also propose a computer program product which has astorage medium on which is stored a computer program which is executableon a computer and executes the design method.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other objects and advantages of the present invention willbecome more apparent and more readily appreciated from the followingdescription of the preferred embodiments, taken in conjunction with theaccompanying drawings of which:

FIG. 1 shows a flowchart of the method according to one embodiment ofthe invention for designing a technical system;

FIG. 2 shows a diagram which illustrates the predictor-corrector methodused in one embodiment of the invention;

FIG. 3 shows a diagram which illustrates the reflection method used inan alternative embodiment of the invention, and

FIG. 4 shows a processor unit for performing the method according to oneembodiment of the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

Reference will now be made in detail to the preferred embodiments of thepresent invention, examples of which are illustrated in the accompanyingdrawings, wherein like reference numerals refer to like elementsthroughout.

A flowchart of a method for designing a technical system is shown inFIG. 1.

First, in step 101, a description form of the technical system isselected. The description form comprises a predetermined number k oftarget functions f=(f1, . . . , fk), with each of the target functionsbeing dependent on n predetermined parameters x₁ to x_(n) of thetechnical system. The target functions are for example the investmentcosts f₁ and the efficiency f₂ of the technical system. In this case thetarget functions are described by the following equation:

${\underset{\_}{f}\left( \underset{\_}{x} \right)} = {\left( \frac{f_{1}\left( \underset{\_}{x} \right)}{f_{2}\left( \underset{\_}{x} \right)} \right) = \left( \frac{{Investment} \cdot {{costs}\left( \underset{\_}{x} \right)}}{{\_ Efficiency}\left( \underset{\_}{x} \right)} \right)}$

where x=(x₁, . . . , x_(n)).

The parameters x₁ to x_(n) can be configuration parameters or operatingparameters of the technical system.

By the method the valid operating points which are used for the designof the technical system are determined by the optimization of the targetfunctions with respect to the parameters, whereby not all targetfunctions f₁ to f_(k) can be optimized simultaneously since theoptimization criteria are generally in competition with one another.

The technical system is further limited in the valid operating points bya predetermined number m of auxiliary conditions h=(h₁(x) . . . , h_(m)(x)) which can be expressed by the following equation:h(x)=0

where 0=(0, . . . , 0) represents a zero vector. In this case what isinvolved is an equality auxiliary condition, with an inequalityauxiliary condition also being able to be considered as an alternative.An inequality condition of this kind is, for example:h(x)<0 or h(x)>0.

In order to solve the optimization problems by inequality auxiliaryconditions, a number m of slack variables s=(s₁, . . . , s_(m)) areintroduced, by which the inequality auxiliary conditions can betransformed into the following equality auxiliary conditions:h(x)+s=0 or h(x)−s=0

With the optimization method described in the present embodiment, thevalid operating points are known as pareto-optimal points, which satisfythe following condition:min _(xεR){ƒ( x )}, R={xε

^(n) |h ( x )=0}

It can be shown that the solutions of this optimization problem are thesolutions of the following nonlinear equation systems:

${F\left( {\underset{\_}{x},\underset{\_}{\lambda},\underset{\_}{a}} \right)} = {\begin{pmatrix}{{\sum\limits_{i = 1}^{k}\;{\alpha_{1} \cdot {\nabla{f_{1}\left( \underset{\_}{x} \right)}}}} + {\sum\limits_{j = 1}^{m}\;{\lambda_{j} \cdot {\nabla{h_{j}\left( \underset{\_}{x} \right)}}}}} \\{\underset{\_}{h}\left( \underset{\_}{x} \right)} \\{{\sum\limits_{l = 1}^{k}\;\alpha_{1}} - 1}\end{pmatrix} = \underset{\_}{0}}$

In this case the auxiliary conditions are taken into account by theLagrange multipliers λ=(λ₁, . . . , λ_(m),) and the target functionsf_(i) are weighted with weighting factors α_(i), whereby care must betaken to ensure that the total of all weighting factors is normalized toone, i.e. Σ_(i=1) ^(k)α₁−1=0. In this case, in particular, theindividual weighting factors α_(i) can also be negative or equal tozero. The solutions of the optimization problem are therefore vectors(x, λ, α) in the (n+m+k)-dimensional variable space of the aboveequation system.

As shown in the Hillemermeier Vol. 110/3 reference, under certainconditions the solutions of this equation system describe a(k−1)-dimensional submanifold M in the variable space.

The below described numerical steps for determining valid operatingpoints are essentially based on the homotopy method described in theHillemermeier Vol. 110/3 reference, wherein a predictor-corrector methodis used for determining pareto-optimal points.

In step 102, proceeding from the description form 101 of the technicalsystem, a first pareto-optimal point z is determined by a standardmethod such as, for example, the weighting method.

In this first pareto-optimal point, in the next step 103, a(k−1)-dimensional tangential plane T_(Z)M to the manifold M of the validoperating points is determined in point z. Toward that end, a Jacobimatrix of the equation system F in point z is subjected to a QRfactorizing. From this, an orthonormal basis {q1 . . . q_(k−1)} is thendetermined which spans the tangential plane. The individual numericalsteps performed in this process are described in detail in theHillemermeier Vol. 110/3 reference.

In the next step 104, a predictor y is determined in this tangentialplane, with the predictor—in contrast to the homotopy method describedin the Hillemermeier Vol. 110/3 reference—being generated by a normallydistributed random number vector b of the dimension k−1 in thetangential plane. In this case the predictor y has the following form:y=z+(q ₁ . . . q _(k−1))b

Through the use of a random number vector such as this, a Brownianmovement can be modeled on the submanifold M, with the Brownian movementbeing able to be represented approximately as follows:dZ_(t) =εP(Z _(t))dB _(t)

where

P(z) is a projection matrix onto the tangential plane T_(Z)M in thevalid operating point z,

ε is a scaling factor, and

B_(t),t ε

₀ ⁺ is a Brownian movement in the variable space.

In order to model this movement, the k−1-dimensional normal distributionN(0_(k−1), tΔεI_(k−1)) is selected for b, where the mean value 0_(k−1)is the (k−1)-dimensional zero vector and the variance is the(k−1)-dimensional identity matrix I_(k−1) multiplied by a step incrementt_(Δ) of the Brownian movement and the scaling factor ε.

An alternative method of determining the predictor is first to determinea normally distributed random number vector in the (m+n+k)-dimensionalvariable space and then to project the vector into the (k−1)-dimensionaltangential plane T_(Z)M.

After this, in step 105, the predictor is projected with the aid of acorrector method, which is, for example, a numerical Newton method, ontothe manifold of the pareto-optimal points. In this way a new validoperating point is determined on the manifold of the pareto-optimalpoints.

The steps 103, 104 and 105 are repeated iteratively, with the operatingpoint determined in the preceding iteration step being used as thestarting point for calculating a new valid operating point.

In step 106 a check is made to determine whether an abort criterion hasbeen met, in other words whether, for example, a predetermined number ofiterations have been performed or a predetermined time limit has beenreached. If this is not the case, a return is made to step 103 and thenext iteration is performed. This is continued until the abort criterionis met.

Once the abort criterion has been met in step 106, in a next step 107the set of determined pareto-optimal points is restricted to thosepoints in which the weighting factors α_(i) are positive.

From these pareto-optimal points, in a final step 108, the user selectsan efficient operating point of the technical system appropriate tohis/her requirements and the technical system is designed using thisefficient operating point.

FIG. 2 shows a two-dimensional graphical representation of thepredictor-corrector method, which may be used in the design method.

In FIG. 2, z^(i) designates a pareto-optimal point on the submanifold M,with this point having been obtained in the i-th iteration step of themethod. In order to determine a new pareto-optimal point, the tangentialplane T_(zi)M to the submanifold M is first determined in the pointz^(i). The tangential plane is indicated by dashed lines in FIG. 2. Inthe next step 104, a predictor point y^(i+1) is then determined usingnormally distributed random numbers in the tangential plane T_(zi)M. Inthe following corrector step 105, which can be, for example, a Newtonmethod, the new pareto-optimal point z^(i+1) is determined. The methodis then continued, with the pareto-optimal point z^(i+1) being used asthe starting point for new predictor step.

FIG. 3 relates to a variation of the method, wherein if predictors withnegative weighting factors α_(i) occur, a reflection is performed inorder to determine a new predictor with positive α_(i). FIG. 3 showsthis projection step being performed in a three-dimensionalrepresentation.

FIG. 3 depicts a case in which, starting from a pareto-optimal point z,a predictor y_(neg) is determined which has a negative α_(i). This isillustrated graphically in that the section between the point z and thepoint y penetrates the tangential plane T_(Z)M in the point S. In thiscase the point S in turn lies on a subplane of the tangential planeT_(Z)M, for the points of which the coordinate α_(i) has the value zero.In order to perform the reflection, the point of intersection S isdetermined first. This can be done using a projection operator whichprojects the α_(i) component from a parameter representation of thestraight line running through the points z and y. After the point S hasbeen determined, the vector xneg running between S and y can now bedetermined. This vector is then dissected into the tangential componentt to the subplane and into a normal component n. Thus, t=x_(neg)−napplies to the tangential component. The reflection step is thenperformed, with the new reflected vector x_(neu) having the sametangential component t as the old vector x_(neg) and the normalcomponent corresponding to the normal component n of the old vectorx_(neg) with the sign reversed. The new vector is thereforex_(neu)=t−n=(x_(neg)−n)−n=X_(neg)−2n. There thus results a new predictory_(neu) which was reflected at the tangential plane T_(Z)M.y_(neu)=S+X_(neu) applies to the new predictor point y_(neu). The abovedescribed reflection method increases the numerical efficiency of themethod since the generation of points with negative weighting factorsα_(i) is avoided and consequently the technical computing resources areused more efficiently.

FIG. 4 shows a processor unit PRZE for performing the method. Theprocessor unit PRZE comprises a processor CPU, a memory MEM and aninput/output interface IOS which is used in a different way via aninterface IFC: An output is made visible on a monitor MON via agraphical interface and/or output on a printer PRT. An input is made viaa mouse MAS or a keyboard TAST. The processor unit PRZE also has a databus BUS which provides the connection from a memory MEM, the processorCPU and the input/output interface IOS. Additional components such as,for example, additional memory, data storage (hard disk) or scanner canalso be connected to the data bus BUS.

The invention has been described in detail with particular reference topreferred embodiments thereof and examples, but it will be understoodthat variations and modifications can be effected within the spirit andscope of the invention covered by the claims which may include thephrase “at least one of A, B and C” or a similar phrase as analternative expression that means one or more of A, B and C may be used,contrary to the holding in Superguide v. DIRECTV, 69 USPQ2d 1865 (Fed.Cir. 2004).

1. A method for designing a technical system having a predetermined setof target functions which are dependent on parameters, comprising:weighting each individual target function with a weighting factor;solving an equation system in a variable space to produce operatingpoints in a solution space, the equation system having the parametersand the weighting factors as variables, the equation system being solvedby a predictor-corrector method comprising: generating a first operatingpoint by determining a predictor as a stochastic variable in thevariable space; and after generating the first operating point,generating a second operating point using a corrector method; and usingthe operating points to design the technical system.
 2. The method asclaimed in claim 1, wherein the predictor is determined by randomnumbers.
 3. The method as claimed in claim 2, wherein the random numbersare normally distributed.
 4. The method as claimed in claim 1, whereinthe stochastic variable relates to a stochastic process Z_(t) whichsatisfies the following equation:dZ _(t) =εP(Z _(t))dB _(t) where P(z) is a projection matrix onto aspace tangential to the solution space in a valid operating point z, εis a scaling factor, and B_(t),tε

₀ ⁺ is a Brownian movement in the variable space.
 5. The method asclaimed in claim 1, wherein pareto-optimal points are determined as theoperating points.
 6. The method as claimed in claim 1, wherein theoperating points are points with positive weighting factors in thesolution space.
 7. The method as claimed in claim 1, wherein theoperating points satisfy one or more auxiliary conditions, with eachauxiliary condition being represented by a further variable of theequation system in the variable space.
 8. The method as claimed in claim7, wherein the auxiliary conditions are equality auxiliary conditionsand/or inequality auxiliary conditions.
 9. The method as claimed inclaim 8, wherein inequality auxiliary conditions are transformed intoequality auxiliary conditions by a slack variable.
 10. The method asclaimed in claim 1, wherein the solution space is a submanifold in thevariable space.
 11. The method as claimed in claim 1, wherein the firstoperating point is generated by a weighting method.
 12. The method asclaimed in claim 1, wherein in generating the first operating point,plane tangential to the solution space is determined and the predictoris determined in said plane.
 13. The method as claimed in claim 1,wherein if a negative predictor associated with a negative weightingfactor occurs, a new predictor is determined by a reflection at asubplane of the solution space having the operating points.
 14. Themethod as claimed in claim 13, wherein a point of intersection of atrajectory which runs between the first operating point and the negativepredictor with the subplane of the solution space is determined; atangential component of a vector spanned by the point of intersectionand the negative predictor is determined at a subplane of the solutionspace, the weighting factor for the negative predictor now being equalto zero; a normal component, associated with the tangential component,of the vector spanned by the point of intersection and the negativepredictor is determined; the new predictor is determined as twice thedifference of the normal component from the negative predictor.
 15. Themethod as claimed in claim 1, wherein the corrector method is aNewtonian method.
 16. The method as claimed in claim 1, wherein theoperating points are determined by iterations of the predictor-correctormethod, with the second operating point of a preceding iteration stepbeing used in a current iteration step as the first operating point ofthe predictor-corrector method.
 17. The method as claimed in claim 16,wherein the iterations are terminated by an abort condition.
 18. Themethod as claimed in claim 17, wherein the abort condition is satisfiedwhen a predetermined number of operating points has been determinedand/or a predetermined time limit has been reached.
 19. A system fordesigning a technical system having a predeterminable set of targetfunctions which are dependent on parameters, comprising; a weightingunit to weight each individual target function with a weighting factor;a processor to solve an equation system having the parameters and theweighting factors as variables in a variable space, the solutions of theequation system forming operating points of a solution space in thevariable space, the operating points being determined by apredictor-corrector method comprising: generating a first operatingpoint by determining a predictor as a stochastic variable in thevariable space; and after generating the first operating point,generating a second operating point in a corrector step; and an outputunit to output the operating points for the design of the technicalsystem.
 20. The system as claimed in claim 19, further comprising arandom number generator for generating the stochastic variable.
 21. Acomputer readable medium on which is stored a computer program toperform a method for designing a technical system having a predeterminedset of target functions which are dependent on parameters, the methodcomprising: weighting each individual target function with a weightingfactor; solving an equation system in a variable space to produceoperating points in a solution space, the equation system having theparameters and the weighting factors as variables, the equation systembeing solved by a predictor-corrector method comprising: generating afirst operating point by determining a predictor as a stochasticvariable in the variable space; and after generating the first operatingpoint, generating a second operating point in a corrector step; andusing the operating points to design the technical system.